An introduction to branes and polytopes

We define an algebraic classification of ½ BPS brane solutions in maximal supergravity theories based on the algebraic structure
provided by the U-duality group. Half-supersymmetric solutions correspond to the longest weights of the U-duality representations
hosting the differential forms coupling to them. This classification is able to capture the differences between standard branes and
non-standard solutions. In particular it reveals a degeneracy for the BPS condition for non-standard solutions, signaling the
possibility to combine them in a bound state preserving the same amount of supersymmetry of the single constituents.

The extensions to non maximal supergravity theories, due to the non-split real form of the U-duality groups, is not immediate and
requires an improvement in the formalism, provided by the theory of real forms of Lie algebras and the Tits-Satake diagrams, to
evaluate the compactness properties of the U-duality charges. We recognize a correspondence between half-supersymmetric solutions and
non-compact longest U-duality charges.

As a relevant application of the algebraic correspondence between ½ BPS branes and non-compact longest weights we briefly discuss the
classification of orbits, under the U-duality group, for single-charge and multi-charge states.

Finally we make a further step in the classification using the Weyl group of the U-duality group to reveal a set of universal rules
fully describing the content of half-supersymmetric solutions in maximal theories. These rules turn out to describe the symmetries of
certain families of uniform polytopes. This allows us to think about ½ BPS branes in maximal supergravities as vertices of certain
families of uniform polytopes.